3.19 \(\int x^4 (d-c^2 d x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=256 \[ -\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^3 (c x-1)^{11/2} (c x+1)^{11/2}}{121 c^5}+\frac{4 b d^3 (c x-1)^{9/2} (c x+1)^{9/2}}{297 c^5}+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{1617 c^5}-\frac{2 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{1925 c^5}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{3465 c^5}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{1155 c^5} \]

[Out]

(-16*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1155*c^5) + (8*b*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(3465*c^5) -
(2*b*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(1925*c^5) + (b*d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(1617*c^5) +
(4*b*d^3*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(297*c^5) + (b*d^3*(-1 + c*x)^(11/2)*(1 + c*x)^(11/2))/(121*c^5) +
(d^3*x^5*(a + b*ArcCosh[c*x]))/5 - (3*c^2*d^3*x^7*(a + b*ArcCosh[c*x]))/7 + (c^4*d^3*x^9*(a + b*ArcCosh[c*x]))
/3 - (c^6*d^3*x^11*(a + b*ArcCosh[c*x]))/11

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Rubi [A]  time = 0.438654, antiderivative size = 326, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {270, 5731, 12, 1610, 1799, 1620} \[ -\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^3 \left (1-c^2 x^2\right )^6}{121 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b d^3 \left (1-c^2 x^2\right )^5}{297 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^3 \left (1-c^2 x^2\right )^4}{1617 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^3 \left (1-c^2 x^2\right )^3}{1925 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{3465 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^3 \left (1-c^2 x^2\right )}{1155 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(16*b*d^3*(1 - c^2*x^2))/(1155*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(3465*c^5*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + (2*b*d^3*(1 - c^2*x^2)^3)/(1925*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x
^2)^4)/(1617*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*d^3*(1 - c^2*x^2)^5)/(297*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + (b*d^3*(1 - c^2*x^2)^6)/(121*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^5*(a + b*ArcCosh[c*x]))/5 - (3*c
^2*d^3*x^7*(a + b*ArcCosh[c*x]))/7 + (c^4*d^3*x^9*(a + b*ArcCosh[c*x]))/3 - (c^6*d^3*x^11*(a + b*ArcCosh[c*x])
)/11

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{1155 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3\right ) \int \frac{x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1155}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{1155 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (231-495 c^2 x+385 c^4 x^2-105 c^6 x^3\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{2310 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{16}{c^4 \sqrt{-1+c^2 x}}-\frac{8 \sqrt{-1+c^2 x}}{c^4}+\frac{6 \left (-1+c^2 x\right )^{3/2}}{c^4}-\frac{5 \left (-1+c^2 x\right )^{5/2}}{c^4}-\frac{140 \left (-1+c^2 x\right )^{7/2}}{c^4}-\frac{105 \left (-1+c^2 x\right )^{9/2}}{c^4}\right ) \, dx,x,x^2\right )}{2310 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{16 b d^3 \left (1-c^2 x^2\right )}{1155 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{3465 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^3 \left (1-c^2 x^2\right )^3}{1925 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \left (1-c^2 x^2\right )^4}{1617 c^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b d^3 \left (1-c^2 x^2\right )^5}{297 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \left (1-c^2 x^2\right )^6}{121 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.260505, size = 147, normalized size = 0.57 \[ -\frac{d^3 \left (3465 a c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (-33075 c^{10} x^{10}+111475 c^8 x^8-117625 c^6 x^6+18933 c^4 x^4+25244 c^2 x^2+50488\right )+3465 b c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right ) \cosh ^{-1}(c x)\right )}{4002075 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

-(d^3*(3465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(50488
+ 25244*c^2*x^2 + 18933*c^4*x^4 - 117625*c^6*x^6 + 111475*c^8*x^8 - 33075*c^10*x^10) + 3465*b*c^5*x^5*(-231 +
495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcCosh[c*x]))/(4002075*c^5)

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Maple [A]  time = 0.018, size = 158, normalized size = 0.6 \begin{align*}{\frac{1}{{c}^{5}} \left ( -{d}^{3}a \left ({\frac{{c}^{11}{x}^{11}}{11}}-{\frac{{c}^{9}{x}^{9}}{3}}+{\frac{3\,{c}^{7}{x}^{7}}{7}}-{\frac{{c}^{5}{x}^{5}}{5}} \right ) -{d}^{3}b \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{11}{x}^{11}}{11}}-{\frac{{\rm arccosh} \left (cx\right ){c}^{9}{x}^{9}}{3}}+{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}-{\frac{{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}-{\frac{33075\,{c}^{10}{x}^{10}-111475\,{c}^{8}{x}^{8}+117625\,{c}^{6}{x}^{6}-18933\,{c}^{4}{x}^{4}-25244\,{c}^{2}{x}^{2}-50488}{4002075}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x)

[Out]

1/c^5*(-d^3*a*(1/11*c^11*x^11-1/3*c^9*x^9+3/7*c^7*x^7-1/5*c^5*x^5)-d^3*b*(1/11*arccosh(c*x)*c^11*x^11-1/3*arcc
osh(c*x)*c^9*x^9+3/7*arccosh(c*x)*c^7*x^7-1/5*arccosh(c*x)*c^5*x^5-1/4002075*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(3307
5*c^10*x^10-111475*c^8*x^8+117625*c^6*x^6-18933*c^4*x^4-25244*c^2*x^2-50488)))

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Maxima [B]  time = 1.25723, size = 628, normalized size = 2.45 \begin{align*} -\frac{1}{11} \, a c^{6} d^{3} x^{11} + \frac{1}{3} \, a c^{4} d^{3} x^{9} - \frac{3}{7} \, a c^{2} d^{3} x^{7} - \frac{1}{7623} \,{\left (693 \, x^{11} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{63 \, \sqrt{c^{2} x^{2} - 1} x^{10}}{c^{2}} + \frac{70 \, \sqrt{c^{2} x^{2} - 1} x^{8}}{c^{4}} + \frac{80 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{6}} + \frac{96 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{10}} + \frac{256 \, \sqrt{c^{2} x^{2} - 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{945} \,{\left (315 \, x^{9} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{5} \, a d^{3} x^{5} - \frac{3}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{75} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 - 3/7*a*c^2*d^3*x^7 - 1/7623*(693*x^11*arccosh(c*x) - (63*sqrt(c^2*x^
2 - 1)*x^10/c^2 + 70*sqrt(c^2*x^2 - 1)*x^8/c^4 + 80*sqrt(c^2*x^2 - 1)*x^6/c^6 + 96*sqrt(c^2*x^2 - 1)*x^4/c^8 +
128*sqrt(c^2*x^2 - 1)*x^2/c^10 + 256*sqrt(c^2*x^2 - 1)/c^12)*c)*b*c^6*d^3 + 1/945*(315*x^9*arccosh(c*x) - (35
*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2 - 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1
)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^10)*c)*b*c^4*d^3 + 1/5*a*d^3*x^5 - 3/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^
2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*
b*c^2*d^3 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^
2*x^2 - 1)/c^6)*c)*b*d^3

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Fricas [A]  time = 1.81655, size = 510, normalized size = 1.99 \begin{align*} -\frac{363825 \, a c^{11} d^{3} x^{11} - 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} - 800415 \, a c^{5} d^{3} x^{5} + 3465 \,{\left (105 \, b c^{11} d^{3} x^{11} - 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} - 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (33075 \, b c^{10} d^{3} x^{10} - 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} - 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} - 50488 \, b d^{3}\right )} \sqrt{c^{2} x^{2} - 1}}{4002075 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/4002075*(363825*a*c^11*d^3*x^11 - 1334025*a*c^9*d^3*x^9 + 1715175*a*c^7*d^3*x^7 - 800415*a*c^5*d^3*x^5 + 34
65*(105*b*c^11*d^3*x^11 - 385*b*c^9*d^3*x^9 + 495*b*c^7*d^3*x^7 - 231*b*c^5*d^3*x^5)*log(c*x + sqrt(c^2*x^2 -
1)) - (33075*b*c^10*d^3*x^10 - 111475*b*c^8*d^3*x^8 + 117625*b*c^6*d^3*x^6 - 18933*b*c^4*d^3*x^4 - 25244*b*c^2
*d^3*x^2 - 50488*b*d^3)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [A]  time = 65.2464, size = 296, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{11}}{11} + \frac{a c^{4} d^{3} x^{9}}{3} - \frac{3 a c^{2} d^{3} x^{7}}{7} + \frac{a d^{3} x^{5}}{5} - \frac{b c^{6} d^{3} x^{11} \operatorname{acosh}{\left (c x \right )}}{11} + \frac{b c^{5} d^{3} x^{10} \sqrt{c^{2} x^{2} - 1}}{121} + \frac{b c^{4} d^{3} x^{9} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{91 b c^{3} d^{3} x^{8} \sqrt{c^{2} x^{2} - 1}}{3267} - \frac{3 b c^{2} d^{3} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{4705 b c d^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{160083} + \frac{b d^{3} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{6311 b d^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{1334025 c} - \frac{25244 b d^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{4002075 c^{3}} - \frac{50488 b d^{3} \sqrt{c^{2} x^{2} - 1}}{4002075 c^{5}} & \text{for}\: c \neq 0 \\\frac{d^{3} x^{5} \left (a + \frac{i \pi b}{2}\right )}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**3*(a+b*acosh(c*x)),x)

[Out]

Piecewise((-a*c**6*d**3*x**11/11 + a*c**4*d**3*x**9/3 - 3*a*c**2*d**3*x**7/7 + a*d**3*x**5/5 - b*c**6*d**3*x**
11*acosh(c*x)/11 + b*c**5*d**3*x**10*sqrt(c**2*x**2 - 1)/121 + b*c**4*d**3*x**9*acosh(c*x)/3 - 91*b*c**3*d**3*
x**8*sqrt(c**2*x**2 - 1)/3267 - 3*b*c**2*d**3*x**7*acosh(c*x)/7 + 4705*b*c*d**3*x**6*sqrt(c**2*x**2 - 1)/16008
3 + b*d**3*x**5*acosh(c*x)/5 - 6311*b*d**3*x**4*sqrt(c**2*x**2 - 1)/(1334025*c) - 25244*b*d**3*x**2*sqrt(c**2*
x**2 - 1)/(4002075*c**3) - 50488*b*d**3*sqrt(c**2*x**2 - 1)/(4002075*c**5), Ne(c, 0)), (d**3*x**5*(a + I*pi*b/
2)/5, True))

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Giac [B]  time = 1.54981, size = 574, normalized size = 2.24 \begin{align*} -\frac{1}{11} \, a c^{6} d^{3} x^{11} + \frac{1}{3} \, a c^{4} d^{3} x^{9} - \frac{3}{7} \, a c^{2} d^{3} x^{7} - \frac{1}{7623} \,{\left (693 \, x^{11} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{63 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{11}{2}} + 385 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 990 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 1386 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 693 \, \sqrt{c^{2} x^{2} - 1}}{c^{11}}\right )} b c^{6} d^{3} + \frac{1}{945} \,{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 180 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 420 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} - 1}}{c^{9}}\right )} b c^{4} d^{3} + \frac{1}{5} \, a d^{3} x^{5} - \frac{3}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b c^{2} d^{3} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

-1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 - 3/7*a*c^2*d^3*x^7 - 1/7623*(693*x^11*log(c*x + sqrt(c^2*x^2 - 1)) -
(63*(c^2*x^2 - 1)^(11/2) + 385*(c^2*x^2 - 1)^(9/2) + 990*(c^2*x^2 - 1)^(7/2) + 1386*(c^2*x^2 - 1)^(5/2) + 115
5*(c^2*x^2 - 1)^(3/2) + 693*sqrt(c^2*x^2 - 1))/c^11)*b*c^6*d^3 + 1/945*(315*x^9*log(c*x + sqrt(c^2*x^2 - 1)) -
(35*(c^2*x^2 - 1)^(9/2) + 180*(c^2*x^2 - 1)^(7/2) + 378*(c^2*x^2 - 1)^(5/2) + 420*(c^2*x^2 - 1)^(3/2) + 315*s
qrt(c^2*x^2 - 1))/c^9)*b*c^4*d^3 + 1/5*a*d^3*x^5 - 3/245*(35*x^7*log(c*x + sqrt(c^2*x^2 - 1)) - (5*(c^2*x^2 -
1)^(7/2) + 21*(c^2*x^2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2 - 1))/c^7)*b*c^2*d^3 + 1/75*(15*x
^5*log(c*x + sqrt(c^2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^2*x^2 - 1)^(3/2) + 15*sqrt(c^2*x^2 - 1))/c^5)
*b*d^3